Cyclic sieving and rational Catalan theory

CYCLIC SIEVING AND RATIONAL CATALAN THEORY

arXiv:1510.08502v1 [math.CO] 28 Oct 2015

MICHELLE BODNAR AND BRENDON RHOADES Abstract. Let a < b be coprime positive integers. Armstrong, Rhoades, and Williams [2] defined a set NC(a, b) of ‘rational noncrossing partitions’, which form a subset of the ordinary noncrossing partitions of {1, 2, . . . , b − 1}. Confirming a conjecture of Armstrong et. al., we prove that NC(a, b) is closed under rotation and prove an instance of the cyclic sieving phenomenon for this rotational action. We also define a rational generalization of the Sa -noncrossing parking functions of Armstrong, Reiner, and Rhoades [1].

1. Introduction This paper is about generalized noncrossing partitions arising in rational Catalan combinatorics. A set partition of [n] := {1, 2, . . . , n} is noncrossing if its blocks do not cross when drawn on a disk whose boundary is labeled clockwise with 1, 2, . . . , n. Noncrossing partitions play a key role in algebraic and geometric combinatorics. Along with an ever-expanding family of other combinatorial objects, the noncrossing partitions of [n] are famously counted by the Catalan number 2n 2n+1 1 1 Cat(n) = n+1 = . n n 2n+1 mn+n+1 1 Given a Fuss parameter m ≥ 1, the Fuss-Catalan number is Cat(m) (n) = mn+n+1 . n When m = 1, this reduces to the classical Catalan number. Many Catalan object have natural FussCatalan generalizations. In the case of noncrossing partitions, the Fuss-Catalan number Cat(m) (n) counts set partitions of [mn] which are m-divisible in the sense that every block has size divisible by m. 1 a+b For coprime positive integers a and b, the rational Catalan number is Cat(a, b) = a+b a,b . Observe that Cat(n, n + 1) = Cat(n) and Cat(n, mn + 1) = Cat(m) (n), so that rational Catalan numbers are a further generalization of Fuss-Catalan numbers. Inspired by favorable representation theoretic properties of the rational Cherednik algebra attached to the symmetric group Sa at parameter ab , the research program of rational Catalan combinatorics seeks to further generalize Catalan combinatorics to the rational setting. Some rational generalizations of Catalan objects have been around for decades – the rational analog of a Dyck path dates back at least to the probability literature of the 1940s. Armstrong, Rhoades, and Williams used rational Dyck paths to define rational analogs of polygon triangulations, noncrossing perfect matchings, and noncrossing partitions [2]. This paper goes deeper into the study of rational noncrossing partitions. For coprime parameters a < b, Armstrong et. al. defined the a, b-noncrossing partitions to be a subset NC(a, b) of the collection of noncrossing partitions of [b − 1] arising from a laser construction involving rational Dyck paths (see Section 2 for details). It was shown that NC(a, b) is counted by Cat(a, b), as it should be, and that NC(n, mn + 1) is the set of m-divisible noncrossing partitions of [mn], as it should be. However, the construction of NC(a, b) in [2] was indirect and involved the intermediate object of rational Dyck paths. This left open the question of whether many of the fundamental properties of classical noncrossing partitions generalize to the rational case. For example, it was unknown whether the set NC(a, b) is closed under the dihedral group of symmetries of the disk with b − 1 Key words and phrases. cyclic sieving, noncrossing partition, rational Catalan theory, parking function. 1

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MICHELLE BODNAR AND BRENDON RHOADES

labeled boundary points. Consequently, the rich theory of counting noncrossing set partitions fixed by a dihedral symmetry (see [12]) lacked a rational extension. Moreover, the unknown status of rotational closure made it difficult to generalize the noncrossing parking functions of Armstrong, Reiner, and Rhoades [1] (or the 2-noncrossing partitions of Edelman [6]) to the rational setting. The core problem was that the natural dihedral symmetries of noncrossing partitions are harder to visualize on the level of Dyck paths, even in the classical case. The purpose of this paper is to resolve the issues in the last paragraph to support NC(a, b) as the ‘correct’ definition of the rational noncrossing partitions. We will prove the following. • NC(a, b) is closed under dihedral symmetries (Corollary 3.16). • The action of rotation on NC(a, b) exhibits a cyclic sieving phenomenon generalizing that for the action of rotation on classical noncrossing partitions (Theorem 5.3). • The numerology of partitions in NC(a, b) with a nontrivial rotational symmetry generalizes that of classical noncrossing partitions with a nontrivial rotational symmetry (Corollaries 4.10, 4.11, 4.12). • Partitions in NC(a, b) can be decorated to obtain a Sa × Zb−1 -set of rational noncrossing parking functions ParkN C (a, b). The formula for the permutation character of this set generalizes the corresponding formula for the classical case (Theorem 6.3). The key to obtaining the rational extensions of classical results presented above will be to develop a better understanding of the set NC(a, b). We will give two new characterizations of this set (Propositions 3.3 and 3.15). The more important of these will involve an idea genuinely new to rational Catalan combinatorics: a new measure of size for blocks of set partitions in NC(a, b) called rank. In the Fuss-Catalan case (a, b) = (n, mn + 1), the rank of a block is determined by its cardinality (Proposition 3.6). However, rank and cardinality diverge at the rational level of generality. The main heuristic of this paper is that: The rank of a block of a rational noncrossing partition is a better measure of its size than its cardinality. In Section 3, we will prove that rank shares additivity and dihedral invariance properties with size. We will prove that ranks (unlike cardinalities) characterize which noncrossing set partitions of [b − 1] lie in NC(a, b). In Section 6, ranks (not cardinalities) will be used to define and study rational noncrossing parking functions. In Sections 4 and 5, ranks (not cardinalities) will be used to give rational analogs of enumerative results for noncrossing partitions with rotational symmetry.

2. Background 2.1. Rational Dyck paths. The prototypical object in rational Catalan combinatorics is the rational Dyck path. Let (a, b) be coprime positive integers. An a, b-Dyck path (or just a Dyck path when a and b are clear from context) is a lattice path in Z2 consisting of north and east steps which starts at (0, 0) ends at (b, a), and stays above the line y = ab x. The 5, 8-Dyck path N EN EEN N EN EEE is shown in Figure 1. When (a, b) = (n, n + 1), rational Dyck paths are equivalent to classical Dyck paths – lattice paths from (0, 0) to (n, n) which stay weakly above y = x. If D is an a, b-Dyck path, a valley of D is a lattice point P on D such that P is immediately preceded by an east step and succeeded by a north step. A vertical run of D is a maximal contiguous sequence of north steps; the number of vertical runs equal the number of valleys. The 5, 8-Dyck path shown in Figure 1 has 4 valleys. The vertical runs of this path have sizes 1, 1, 2, and 1. The numerology associated to rational Dyck paths generalizes that of classical Dyck paths. The 1 a+b number of a, b-Dyck paths is the rational Catalan number Cat(a, b) = a+b . The set of a, b-Dyck a,b


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Figure 1. A 5, 8-Dyck path and the corresponding 5, 8-noncrossing partition. The visibility bijection is shown with colors. paths D with k vertical runs is counted by the rational Narayana number b−1 1 a . (1) Nar(a, b; k) := a k k−1 P Given a length a vector of nonnegative integers r = (r1 , r2 , . . . , ra ) satisfying iri = a, the number of a, b-Dyck paths with ri vertical runs of size i for 1 ≤ i ≤ a is given by the rational Kreweras number 1 b (b − 1)! (2) Krew(a, b; r) := = , b r1 , r2 , . . . , ra , b − k r1 !r2 ! · · · ra !(b − k)! P where k = ri is the total number of vertical runs. For example, the 5, 7-Dyck path shown in Figure 2 contributes to Krew(5, 7; r), where r = (1, 2, 0, 0, 0). 2.2. Noncrossing partitions. A set partition π of [n] is called noncrossing if, for all indices 1 ≤ i < j < k < ` ≤ n, we have that i ∼ k in π and j ∼ ` ∈ π together imply that i ∼ j ∼ k ∼ ` in π. Equivalently, the set partition π is noncrossing if and only if the convex hulls of the blocks of π do not intersect when drawn on the disk with boundary points labeled clockwise with 1, 2, . . . , n. We let NC(n) denote the collection of noncrossing partitions of [n]. The rotation operator [n] rot acts on the indexset [n], the power set 2 , and the collection NC(n) by the permutation 1 2 ... n − 1 n . These three sets also carry an action of the reflection operator rfn by 2 3 ... n 1 1 2 ... n − 1 n the permutation . Together, rot and rfn generate a dihedral action n n − 1 ... 2 1 on these sets. 2.3. Rational noncrossing partitions. In [2], rational Dyck paths were used to construct a rational generalization of the noncrossing partitions. Let D be an a, b-Dyck path and let P 6= (0, 0) be a lattice point which is at the bottom of a north step of D. The laser `(P ) is the line segment of slope ab which is ‘fired’ northeast from P and continues until it intersects the Dyck path D. By coprimality, the east endpoint of `(P ) is necessarily on the interior of an east step of D. Let D be an a, b-Dyck path. We define a set partition π(D) of [b − 1] as follows. Label the east ends of the non-terminal east steps of D from left to right with 1, 2, . . . , b − 1 and fire lasers from all of the valleys of D. The set partition π(D) is defined by the ‘visibility’ relation i ∼ j if and only if the labels i and j are not separated by laser fire. (Here we consider labels to lie slightly below their lattice points.) By construction, the set partition π(D) is noncrossing. An example of this construction when (a, b) = (5, 8) is shown in Figure 1. If D is an a, b-Dyck path, we have a natural ‘visibility’ bijection from the set of vertical runs of D to the set of blocks

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Figure 2. A 5, 7-Dyck path and the corresponding 5, 7-homogeneous noncrossing partition. of π(D) which associates a vertical run at y = i to the block of π(D) whose minimum element is i + 1. The visibility bijection is shown with colors in Figure 1. It will be convenient to think of the lasers fired from the valleys of a Dyck path D in terms of their endpoints. We let the laser set `(D) of the a, b-Dyck path D be the set of pairs (i, j) such that D contains a laser starting from a valley with x-coordinate i and ending in the interior of an east step with west x-coordinate j. For the 5, 8-Dyck path D shown in Figure 1, we have that `(D) = {(1, 2), (3, 7), (4, 5)}. For a < b coprime, we define the set of admissible lasers rb for some r = 1, 2, . . . , a − 1 . A(a, b) := (i, j) : 1 ≤ i < j ≤ b − 1 and j − i = a Slope considerations show that (i, j) ∈ `(D) for some a, b-Dyck path D if and only if (i, j) ∈ A(a, b). By considering π(D) for all possible a, b-Dyck paths D, we get the set of a, b-noncrossing partitions NC(a, b) := {π(D) : D an a, b-Dyck path} (This is called the set of inhomogeneous a, b-noncrossing partitions in [2].) It is clear from construction that NC(a, b) ⊆ NC(b − 1). Some basic facts about a, b-noncrossing partitions are as follows. Proposition 2.1. Let a < b be coprime positive integers. (1) The map π : {a, b-Dyck paths} → NC(a, b) is bijective, so that |NC(a, b)| = Cat(a, b) and the number of a, b-noncrossing partitions with k blocks is the rational Narayana number Nar(a, b; k). (2) If π ∈ NC(a, b) and π 0 is a noncrossing partition of [b − 1] which coarsens π, then π 0 ∈ NC(a, b). When (a, b) = (n, n + 1), the set NC(n, n + 1) of rational noncrossing partitions is just the set NC(n) of all noncrossing partitions of [n]. When (a, b) = (n, mn + 1) we have that NC(n, mn + 1) of rational noncrossing partitions is the set of all noncrossing partitions of [mn] which are m-divisible in the sense that every block size is divisible by m. Armstrong, Rhoades, and Williams posed the problem of finding an analogous ‘intrinsic’ characterization of NC(a, b) for arbitrary a < b coprime. We give two such characterizations in Section 3. 2.4. Homogeneous rational noncrossing partitions. Rational Dyck paths are used in [2] to construct a generalization of noncrossing perfect matchings on the set [2n]. The construction is similar to that of the rational noncrossing partitions. Let (a, b) be coprime and let D be an a, b-Dyck path. We construct a set partition π(D) of the set [a + b − 1] as follows. Label the interior lattice points of D from southwest to northeast with


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1, 2, . . . , a + b − 1. Fire lasers of slope ab from every lattice point of D which is at the south end of a north step (not just those lattice points which are valleys of D). The set partition π(D) is given by declaring i ∼ j if and only if the labels i and j are not separated by laser fire. As before, we consider labels to be slightly below their lattice points. Topological considerations make it clear that π(D) is a noncrossing partition of [a + b − 1]. An example of this construction is shown in Figure 2 for (a, b) = (5, 8). Considering π(D) for all possible a, b-Dyck paths D gives rise to the set of a, b-homogeneous rational noncrossing partitions HNC(a, b) := {π(D) : D an a, b-Dyck path}. The adjective ‘homogeneous’ refers to the fact that every set partition in HNC(a, b) has a blocks. By construction, we have that HNC(a, b) ⊆ NC(a + b − 1). When (a, b) = (n, n + 1), the set HNC(n, n + 1) is the set of noncrossing perfect matchings on [2n]. When (a, b) = (n, mn + 1), the set HNC(n, mn + 1) is the set of noncrossing set partitions on [mn] in which every block has size m (these are also called m-equal noncrossing partitions). Some basic facts about HNC(a, b) for general a < b are as follows. Proposition 2.2. Let a < b be coprime positive integers. (1) The map π : {a, b-Dyck paths} → HNC(a, b) is bijective, so that |HNC(a, b)| = Cat(a, b). (2) The set HNC(a, b) is closed under the rotation operator rot. In [2] a promotion operator on a, b-Dyck paths is shown to intertwine the action of rot with the map π from paths to homogeneous noncrossing partitions. We will give an inhomogeneous analog of this promotion operator in the next section. 3. Characterizations of Rational Noncrossing Partitions 3.1. The action of rotation. Let a < b be coprime. In this subsection we will prove that NC(a, b) is closed under the rotation operator rot by describing rotation (or, rather, its inverse) as an operator on a, b-Dyck paths. We will define an operator rot0 : {a, b-Dyck paths} −→ {a, b-Dyck paths} on the set of a, b-Dyck paths which satisfies rot−1 ◦ π = π ◦ rot0 . The definition of rot0 is probably best understood visually. Figure 3 shows three 11, 13-Dyck paths. The middle path is the image of the left path under rot0 and the right path is the image of the middle path under rot0 . The valley lasers are fired on each of these Dyck paths, and the corresponding partitions in NC(11, 13) are shown below, but the vertex labels on the Dyck paths are omitted for legibility. Let D1 be the 11, 13-Dyck path on the left of Figure 3. The westernmost horizontal run of D1 has size > 1. Because of this, we define D2 := rot0 (D1 ) by removing one step from this horizontal run and adding it to the easternmost horizontal run of D1 . The lattice path D2 is displayed in the middle of Figure 3. Informally, the path D2 is obtained from D1 by translating one unit west. Let D2 be the 11, 13-Dyck path in the middle of Figure 3. The westernmost horizontal run of D2 has size 1. The definition of D3 := rot0 (D2 ) is more complicated in this case. We break the lattice path D2 up into four subpaths. The first subpath, shown in black, is the initial northern run of D2 . The second subpath, shown in red, is the single east step which occurs after this run. The third subpath, shown in green, extends from the westernmost valley of D2 to the (necessarily east) step just before the laser fired from this valley hits D2 . The fourth subpath, shown in blue, extends from the end of the third subpath to the terminal point (b, a) of D2 . The path D3 is obtained by

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Figure 3. The action of rot0 on a, b-Dyck paths. concatenating the third, first, fourth, and second subpaths, in that order. The concatenation of green, then black, then blue, then red is shown on the right of Figure 3. More formally, given an a, b-Dyck path D, the definition of rot0 (D) breaks up into three cases. Let D = N i1 E j1 · · · N im E jm be the decomposition of D into nonempty vertical and horizontal runs. (1) If m = 1 so that D = N a E b , we set rot0 (D) := N a E b = D. (2) If m, j1 > 1, we set rot0 (D) := N i1 E j1 −1 N i2 E j2 · · · N im E jm +1 . (3) If m > 1 and j1 = 1, let P = (1, i1 ) be the westernmost valley of D. The laser `(P ) fired from P hits D on a horizontal run E ik for some 2 ≤ k ≤ m. Suppose that `(P ) hits the horizontal run E ik on step r, where 2 ≤ r ≤ ik . We set rot0 (D) := N i2 E i2 · · · N ik−1 E ik−1 N ik E r−1 N i1 E ik −r+1 N ik+1 E ik+1 · · · N im E jm +1 . Proposition 3.1. The definition of rot0 given above gives a well defined operator on the set of a, b-Dyck paths. As operators {a, b-Dyck paths } −→ NC(a, b), we have that rot−1 ◦ π = π ◦ rot0 . In particular, the set NC(a, b) is closed under rotation. Proof. Let D be an a, b-Dyck path. If the westernmost horizontal run of D has size > 1, it is clear that rot0 (D) is also an a, b-Dyck path and that the corresponding set partitions are related by rot−1 (π(D)) = π(rot0 (D)). We therefore assume that the westernmost horizontal run of D consists of a single step. We claim that the lattice path rot0 (D) stays above the diagonal y = ab x. Indeed, consider the decomposition of rot0 (D) as in Figure 3. The first (green) subpath of rot0 (D) stays above y = ab x because the laser fired from the westernmost valley of D has slope ab . The second (black) subpath of rot0 (D) is a vertical run, so the concatenation of the first and second subpaths of rot0 (D) stay above y = ab x. The third (blue) subpath of rot0 (D) is just the corresponding subpath of D translated one unit west, and certainly stays above the line y = ab x. Since the fourth (red) subpath of rot0 (D) is a single east step, we conclude that the entire path rot0 (D) stays above y = ab x, and so is an a, b-Dyck path.


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Figure 4. An example of Kreweras complement. By the last paragraph, the set partition π(rot0 (D)) ∈ NC(a, b) is well defined. We argue that rot−1 (π(D)) = π(rot0 (D)). To do this, we consider how the valley lasers of rot0 (D) relate to the corresponding valley lasers of D. • The valley lasers in the third (blue) subpath of rot0 (D) are just the valley lasers in the corresponding subpath of D shifted one unit west. • The valley lasers in the first (green) subpath of rot0 (D) are either the valley lasers in the corresponding subpath of D shifted one unit west, or hit rot0 (D) on its terminal east step, depending on whether these lasers hit D in its third (green) or fourth (blue) subpaths. • The valley laser of the westernmost valley in D is replaced by the laser in the valley of rot0 (D) between its first (green) and second (black) subpaths, which necessarily hits rot0 (D) on its terminal east step. From this description of the valley lasers of rot0 (D), one checks that rot−1 (π(D)) = π(rot0 (D)), as desired. 3.2. Characterization from Kreweras complement. Our first characterization of rational noncrossing partitions gives a description of their Kreweras complements. Let π be a noncrossing partition of [n]. The Kreweras complement krew(π) is the noncrossing partition obtained by drawing the 2n vertices 1, 10 , 2, 20 , . . . , n, n0 clockwise on the boundary of a disk in that order, drawing the blocks of π on the unprimed vertices, and letting krew(π) be the unique coarsest partition of the primed vertices which introduces no crossings. The map krew : NC(n) → NC(n) satisfies krew2 = rot, and so is a bijection. An example of Kreweras complementation for n = 7 is shown in Figure 4. We have that krew : {{1, 2}, {3, 6}, {4, 5}, {7}} 7→ {{1}, {2, 6, 7}, {3, 5}, {4}}. For (a, b) 6= (n, n+1), the set NC(a, b) is not closed under Kreweras complement. Indeed, the one block set partition {[b − 1]} is contained in NC(a, b) but its Kreweras complement (the all singletons set partition) is not. On the other hand, the set NC(a, b) is in bijective correspondence with its Kreweras image krew(NC(a, b)). The problem of characterizing NC(a, b) is therefore equivalent to the problem of characterizing its Kreweras image. Given π ∈ NC(a, b), there is a simple relationship between the blocks of krew(π) ∈ NC(b − 1) and the laser set `(D) of the a, b-Dyck path D corresponding to π. Lemma 3.2. Let a < b be coprime, let π ∈ NC(a, b) have corresponding a, b-Dyck path D, and let krew(π) be the Kreweras complement of π. The laser set `(D) is `(D) = {(i, max(B)) : B is a block of krew(π) and i is a nonmaximal element of B}. Proof. This is clear from the topological definition of π(D) and Kreweras complement.

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Let π ∈ NC(b − 1) be a noncrossing partition and suppose we would like to determine whether π is in NC(a, b). In light of Lemma 3.2, it is enough to know whether there exists an a, b-Dyck path D whose laser set consists of pairs (i, max(B)), where i ∈ B runs over all nonmaximal elements of all blocks B of krew(π). A characterization of the laser sets of a, b-Dyck paths can be read off from the results of [13]. In [13] this characterization was used to prove that the rational analog of the associahedron is a flag simplicial complex. Proposition 3.3. Let a < b be coprime and let π be a noncrossing partition of [b − 1]. We have that π is an a, b-noncrossing partition if and only if for every block B of the Kreweras complement krew(π) we have that • (i, max(B)) ∈ A(a, b) for every nonmaximal element i ∈ B, and • for any two nonmaximal elements i < j in B, l am a am l > (j − i) . (max(B) − i) − (max(B) − j) b b b Proof. As mentioned above, we know that π ∈ NC(a, b) if and only if there exists an a, b-Dyck path D whose laser set consists of all possible pairs (i, max(B)), where B ranges over all blocks of krew(π) and i ranges over all nonmaximal elements of B. For such an a, b-Dyck path D to exist, it is certainly necessary that (i, max(B)) ∈ A(a, b) always, so we assume this condition holds. By [13, Proposition 1.3] and [13, Lemma 4.3], we know that an a, b-Dyck path D as in the previous paragraph exists if and only if for every two pairs (i, max(B)) and (i0 , max(B 0 )), there exists an a, b-Dyck path D with laser set `(D) = {(i, max(B)), (j, max(B 0 ))}. By [13, Lemma 5.3] and [13, Proposition 5.5], only if max(B) = max(B 0 ) (so that such an a, b-Dyck path D does anot exist if and a a 0 B = B ) and (max(B) − i) b − (max(B) −j) b < (j − i) b , where i < j. Since 1 ≤ i < j < b − 1, we have (max(B) − i) ab − (max(B) − j) ab 6= (j − i) ab and the result follows. Example 3.4. Let (a, b) = (5, 8) and consider the following three partitions in NC(7): π1 := {{1}, {2, 6, 7}, {3}, {4, 5}}, π2 := {{1}, {2, 6, 7}, {3, 4}, {5}}, π3 := {{1}, {2, 7}, {3, 4, 5}, {6}}. To determine which of π1 , π2 , π3 belong to NC(5, 8), we start by computing their Kreweras images: krew(π1 ) := {{1, 7}, {2, 3, 5}, {4}, {6}}, krew(π2 ) := {{1, 7}, {2, 4, 5}, {3}, {6}}, krew(π3 ) := {{1, 7}, {2, 5, 6}, {3}, {4}}. Since {2, 3, 5} is a block of krew(π1 )and (3, 5) / A(5, 8), we / NC(5, 8). Since ∈ conclude that π1 ∈ / {2, 4, 5} is a block of krew(π2 ) and (5 − 2) 58 − (5 − 4) 58 ≤ (4 − 2) 58 , we conclude that π2 ∈ NC(5, 8). Since both of the bullets in Proposition 3.3 hold for π3 , we conclude that π3 ∈ NC(5, 8). 3.3. a, b-ranks. In order to state our second characterization of a, b-noncrossing partitions, we introduce a new way to measure the ‘size’ of a block B of a noncrossing partition π other than its cardinality |B|. Our rational analog of block size is as follows. Definition 3.5. Let a < b be coprime positive integers and let π ∈ NC(b − 1) be a noncrossing partition of [b − 1]. We assign an integer rankπa,b (B) ∈ Z to every block B of π (or just rank(B) when a, b, and π are clear from context) by the following recursive procedure. Let be the partial order on the blocks of π defined by B 0 B ⇐⇒ [min(B 0 ), max(B 0 )] ⊆ [min(B), max(B)].


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The integers rankπa,b (B) are implicitly determined by the formula l X am (3) rankπa,b (B 0 ) = (max(B) − min(B) + 1) , b 0 B B

for all blocks B ∈ π. While we define rankπa,b (·) as a function on the blocks of an arbitrary noncrossing partition of [b − 1], this notion of rank will be more useful for a, b-noncrossing partitions. We will see that rank is more useful than cardinality as a block size measure in rational Catalan theory. This subsection proves basic properties of the function rankπa,b (·) on the blocks of a, b-noncrossing partitions. In the Fuss-Catalan case b ≡ 1 (mod a), rank and cardinality are equivalent. Proposition 3.6. Let (a, b) = (n, mn + 1), let π ∈ NC(n, mn + 1) be an m-divisible noncrossing partition of [mn], and let B be a block of π. We have that |B| . m Proof. For any block B of π, we have that 1 ≤ min(B) ≤ max(B) ≤ mn and the divisibility relation m|(max(B) − min(B) + 1), so that n max(B) − min(B) + 1 (max(B) − min(B) + 1) = . mn + 1 m rankπn,mn+1 (B) =

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|B| m

therefore satisfies the recursion for rankπn,mn+1 (·).

For general a < b coprime, the a, b-rank of a block of an a, b-noncrossing partition π ∈ NC(a, b) is not necessarily determined by its cardinality. For example, consider (a, b) = (3, 5). Then π = {{1, 3}, {2}, {4}} ∈ NC(3, 5). We have that rankπ3,5 ({1, 3}) = rankπ3,5 ({2}) = rankπ3,5 ({4}) = 1. Proposition 3.6 shows that the divergence between rank and cardinality is a genuinely new feature of rational Catalan combinatorics and is invisible at the Catalan and Fuss-Catalan levels of generality. Definition 3.7. If π ∈ NC(a, b) is an a, b-noncrossing partition, the rank sequence R(π) = (r1 , . . . , rb−1 ) of π is the sequence of nonnegative integers given by ( rank(B) if i = min(B) for some block B of π, ri := 0 otherwise. For example, the rank sequence of the 5, 8-noncrossing partition shown in Figure 1 has rank sequence (1, 1, 0, 2, 1, 0, 0). Observe that this is also the sequence of vertical runs of the corresponding 5, 8-Dyck path, so that the rank sequence is equivalent to the Dyck path. This is a general phenomenon. Proposition 3.8. Let a < b be coprime, let π be an a, b-noncrossing partition with rank sequence R(π) = (r1 , . . . , rb−1 ), and let D be the a, b-Dyck path associated to π. (1) For any block B of π, the vertical run of D visible from π has size rank(B). (2) The Dyck path D is given by D = N r1 EN r2 E · · · N rb−1 EE. Proof. For any block B of π, we have that min(B) labels the lattice point just to the right of the vertical run labeled by B. Therefore, (2) follows from (1). To prove (1), recall the partial order on the blocks of π. If B is minimal with respect to , then B = [min(B), max(B)] is an interval. The labels of B must appear on a single horizontal run

10


of D, say on the line y = c. Let iB be the length of the vertical run visible from B, let P be the valley at the bottom of this vertical run, and let `(P ) be the laser fired from P . The laser `(P ) must intersect the line y = c. We claim that it does so in the open x-interval (max(B), max(B) + 1). Indeed, by coprimality there exists a unique lattice point Q on D which is northwest of the laser `(P ) and has minimum horizontal distance to the laser `(P ). Let m be the x-coordinate of Q. We claim that m ∈ B. Indeed, if m ∈ / B, there must be a laser `(Q0 ) fired from 0 0 a valley Q which separates m from B. But then Q would be closer than Q to `(P ), so we conclude that m ∈ B. Since B = [min(B), max(B)] is an interval, we have that `(P ) intersects y = c in the open x-interval (max(B), max(B) + 1). This implies that rank(B) = iB , as desired. Now suppose that B is not -minimal among the blocks of π. Then the interval [min(B), max(B)] is a union of at least two blocks of π. For any block B 0 contained in this interval, let iB 0 denote the size of the vertical run visible from B 0 . We may inductively assume that, for all B 0 6= B, we have that rankπa,b (B 0 ) = iB 0 . Let π0 be the set partition obtained from π by merging the blocks contained in [min(B), max(B)]. Then π0 is noncrossing, and hence a, b-noncrossing by Proposition 2.1 (2). The recursion for rank says that X X 0 iB 0 , rankπa,b ([min(B), max(B)]) = rankπa,b (B) = rankπa,b (B) + B 0 ⊆[min(B),max(B)]

B 0 ⊆[min(B),max(B)] B 0 6=B

where the second equality used the inductive hypothesis. Moreover, the Dyck path D0 corresponding to π0 is obtained from D by replacing the portion between the x-coordinates min(B) and max(B) with a single vertical run, followed by a single P horizontal run. In particular, the sizeπof the vertical run in D0 visible from [min(B), max(B)] is B 0 ⊆[min(B),max(B)] iB 0 . This forces ranka,b (B) = iB , as desired. The cardinality function | · | on blocks of noncrossing partitions satisfies the following two properties. • Let π and π 0 be noncrossing partitions such that π refines π 0 . For any block B 0 of π 0 , write B 0 = B1 ] · · · ] Bk , where B1 , . . . , Bk are blocks of π. We have |B 0 | = |B1 | + · · · + |Bk |. • The function | · | on blocks of noncrossing set partitions is invariant under the dihedral action of hrot, rfni. To justify our intuition that rank measures size, we prove that rankπa,b (·) enjoys the same properties on the set of a, b-noncrossing partitions. Proposition 3.9. Let a < b be coprime, let π, π 0 ∈ NC(a, b), and assume that π refines π 0 . Let B 0 be a block of π 0 and let B1 , . . . , Bk be the blocks of π such that B 0 = B1 ] · · · ] Bk . We have 0

rankπa,b (B 0 ) = rankπa,b (B1 ) + · · · + rankπa,b (Bk ). Proof. It suffices to consider the case where π 0 is obtained by merging two blocks of π, say B1 and B2 . Suppose min(B1 ) < min(B2 ). If D, D0 are the a, b-Dyck paths corresponding to π, π 0 , then D0 is obtained from D by moving the vertical run visible from B2 on top of the vertical run visible from B1 . The merged vertical run in D0 is visible from B1 ] B2 . The result follows from Proposition 3.8. Proposition 3.9 fails for partitions π, π ∈ NC(b − 1) which are not both a, b-noncrossing. For example, let (a, b) = (2, 5), π = {{1, 2, 3}, {4}}, and π 0 = {{1, 2, 3, 4}}. We have rankπ2,5 ({1, 2, 3}) = 0 2, rankπ2,5 ({4}) = 1, and rankπ2,5 ({1, 2, 3, 4}) = 2.


11

Proposition 3.10. Let a < b be coprime, let π ∈ NC(a, b), and let B be a block of π. We have rot(π)

(rot(B)),

rfn(π)

(rfn(B)).

(4)

rankπa,b (B) = ranka,b

(5)


Proof. Since the intervals [min(B), max(B)] and [min(rfn(B)), max(rfn(B))] have the same length, reflection invariance is clear (and still holds if π fails to be a, b-noncrossing). Recall that rot acts by adding 1 to every index, modulo b − 1. Rotation invariance is therefore clear unless B has the form B = {i1 < · · · < ij < b − 1} with i1 > 1. In this case, the intervals [1, i1 − 1], [i2 + 1, i3 − 1], . . . , [ij + 1, b − 2] must each be unions of blocks of π. Let B1 , . . . , Br be rot(π) a complete list of these remaining blocks. We certainly have rankπa,b (Bi ) = ranka,b (rot(Bi )) for 1 ≤ i ≤ r. On the other hand, Proposition 3.1 and the assumption that π ∈ NC(a, b) guarantee that rot(π) ∈ NC(a, b). Proposition 3.8 shows that a = rankπa,b (B) + rankπa,b (B1 ) + · · · + rankπa,b (Br ) rot(π)

= ranka,b

rot(π)

(rot(B)) + ranka,b

rot(π)

Since rankπa,b (Bi ) = ranka,b

rot(π)

(rot(B1 )) + · · · + ranka,b

(rot(Br )). rot(π)

(rot(Bi )) for 1 ≤ i ≤ r, we get rankπa,b (B) = ranka,b

(rot(B)).

Proposition 3.10 does not hold for partitions π ∈ NC(b − 1) which are not a, b-noncrossing. For example, take (a, b) = (2, 5) and π = {{1}, {2, 3, 4}}, so that rot(π) = {{1, 3, 4}{2}}. We have rot(π) that rankπ2,5 ({2, 3, 4}) = 2 but rank2,5 ({1, 3, 4}) = 1. 3.4. Characterization from rank function. The a, b-rank function rankπa,b (·) can be used to decide whether a given noncrossing partition π ∈ NC(b − 1) is a, b-noncrossing. This is our first application of a, b-rank to rational Catalan theory. To begin, we introduce the notion of a valid a, b-ranking. Definition 3.11. Let π ∈ NC(b − 1) be an arbitrary noncrossing partition of [b − 1]. We say that π has a valid a, b-ranking if rankπa,b (B) > 0 for every block B ∈ π and X rank(B) = a. B∈π

Example 3.12. Let (a, b) = (3, 5) and consider the three noncrossing partitions π1 = {{1, 2, 4}, {3}}, π2 = {{1, 4}, {2}, {3}}, and π3 = {{1, 2}, {3}, {4}}. 2 We have that π1 has a valid 3, 5-ranking, but π2 and π3 do not. Indeed, we have rankπ3,5 ({1, 4}) = 0 π3 π3 π3 and rank3,5 ({1, 2}) + rank3,5 ({3}) + rank3,5 ({4}) = 2 + 1 + 1 > 3. We record the following lemma for future use. Lemma 3.13. Let a < b be coprime and let π ∈ N C(b − 1) be an arbitrary noncrossing partition of [b − 1]. We have X (6) rank(B) ≥ a. B∈π

Proof. Recall the partial order on the blocks of π. If B 0 is maximal with respect to , we have l X a am > (max(B 0 ) − min(B 0 ) + 1) . rank(B) = (max(B 0 ) − min(B 0 ) + 1) b b 0 BB

Summing over all -maximal blocks B 0 gives X a (7) rank(B) > (b − 1) , b B∈π

12


which is equivalent to the desired inequality since

P

B∈π

rank(B) ∈ Z.

By Proposition 3.8, every a, b-noncrossing partition has a valid a, b-ranking. It would be nice if the converse held, but it does not; if (a, b) = (2, 5), the partition π = {{1, 2, 4}, {3}} has a valid 2, 5-ranking but π ∈ / NC(2, 5). Our second characterization of a, b-noncrossing partitions states that π ∈ NC(a, b) if and only if every element in the orbit of π under rotation has a valid a, b-ranking. To prove this statement, we start by examining how validity behaves under coarsening. In general, having a valid a, b-ranking is not closed under coarsening of noncrossing partitions. For example, let (a, b) = (5, 11) and consider the two partitions π, π 0 ∈ NC(10) given by π = {{1, 6, 7, 8, 9, 10}, {2}, {3}, {4}, {5}}, π 0 = {{1, 6, 7, 8, 9, 10}, {2, 5}, {3}, {4}}. Then π has a valid 5, 11-ranking but π 0 does not. On the other hand, certain types of coarsening do preserve validity. Lemma 3.14. Let a < b be coprime and let π ∈ NC(b − 1) be a noncrossing partition such that π has a valid a, b-ranking. Suppose that π 0 ∈ NC(b − 1) is another noncrossing partition obtained from π by one of the following two operations: (1) merging two blocks B0 , B1 of π with min(B1 ) = max(B0 ) + 1, or (2) merging two blocks B0 , B1 of π with B0 ≺ B1 . The noncrossing partition π 0 has a valid a, b-ranking. Observe that if π, π 0 ∈ NC(b − 1) are such that π refines π 0 and π 0 is a union of intervals, then π 0 can be obtained from π by a sequence of coarsenings as described in Lemma 3.14. Proof. First assume that π 0 is obtained from π by replacing B0 and B1 with B0 ∪B1 as in Condition 1. For i = 0, 1, let ri denote the sum of the ranks of the blocks B of π satisfying B ≺ Bi . By the definition of rank, we have l am (8) rankπa,b (Bi ) + ri = (max(Bi ) − min(Bi ) + 1) b for i = 0, 1. Adding these two equations together and recalling that dxe+dye−1 ≤ dx+ye ≤ dxe+dye for any x, y ∈ R, we get (9)

0

rankπa,b (B0 ) + rankπa,b (B1 ) − 1 ≤ rankπa,b (B0 ∪ B1 ) ≤ rankπa,b (B0 ) + rankπa,b (B1 ). 0

Since π has a valid a, b-ranking, we have rankπa,b (Bi ) > 0 for i = 0, 1, so that rankπa,b (B0 ∪ B1 ) > 0. Our analysis breaks up into two cases depending on whether B0 ∪ B1 is a -maximal block of π 0 . If B0 ∪ B1 is a -maximal block of π 0 , for every block B of π 0 with B 6= B0 , B1 we have P 0 0 rankπa,b (B) = rankπa,b (B), so that the chain of inequalities in (9) implies B∈π0 rankπa,b (B) ≤ a, so P π0 0 B∈π 0 ranka,b (B) = a by Lemma 3.13. Since every block of π has a positive rank, we conclude that π 0 has a valid a, b-ranking. If B0 ∪ B1 is not a -maximal block of π 0 , there exists a unique block B2 ∈ π 0 such that B2 0 covers B0 ∪ B1 in . The recursion for rankπa,b (·) gives l am 0 0 (10) rankπa,b (B0 ∪ B1 ) + r0 + r1 + r2 + rankπa,b (B2 ) = (max(B2 ) − min(B2 ) + 1) , b 0 where r2 is the sum of the ranks of all of the blocks B of π satisfying B ≺ B2 but B 6 B0 , B1 . 0 Combining (10) with the inequalities in (9), we see that rankπa,b (B2 ) ≥ rankπa,b (B2 ) > 0. Since l X X am 0 = rankπa,b (B 0 ), (11) rankπa,b (B) = (max(B2 ) − min(B2 ) + 1) b 0 0 B∈π BB2

B ∈π B 0 B2


13

0

we have rankπa,b (B) = rankπa,b (B) for all blocks B 6= B0 , B1 , B2 . In particular, the rank of every P 0 block of π 0 is positive and B∈π0 rankπa,b (B) = a. We conclude that π 0 has a valid a, b-ranking. Now assume that π 0 is obtained from π by replacing B0 and B1 with B0 ∪ B1 as in Condition 2. It follows that B1 covers B0 in the -order. By the recursion for rank, (12)

0

rankπa,b (B0 ∪ B1 ) = rankπa,b (B0 ) + rankπa,b (B1 )

and the ranks of all other blocks of π 0 equal the corresponding ranks in π 0 , so that π 0 has a valid a, b-ranking. We are ready to prove our a, b-rank characterization of a, b-noncrossing partitions. Proposition 3.15. Let a < b be coprime and let π be a noncrossing partition of [b − 1]. We have that π is an a, b-noncrossing partition if and only if every partition in the orbit of π under rotation has a valid a, b-ranking. Proof. Suppose π ∈ NC(a, b). By Proposition 3.1, we know that the rotation orbit of π is contained in NC(a, b), so that every partition in this orbit has a valid a, b-ranking. For the converse, suppose that π ∈ NC(b − 1) − NC(a, b). We argue that some partition in the rotation orbit of π does not have a valid (a, b)-ranking. Consider the Kreweras complement krew(π). If there is a block B ∈ krew(π) and an index i ∈ B − {max(B)} such that (i, max(B)) ∈ / A(a, b), then π refines the two-block set partition π 0 := {[i+1, max(B)], [b−1]−[i+1, max(B)]} and rot−i (π) refines rot−i (π 0 ) = {[1, max(B)−i], [max(B)− i + 1, b − 1]}. Since rot−i (π) consists of two intervals, we have that rot−i (π 0 ) can be obtained from rot−i (π) by a sequence of coarsenings as in Lemma 3.14. The condition (i, max(B)) ∈ / A(a, b) means that l am l am (max(B) − i) + (b − max(B) + i − 1) > a, b b so that rot−i (π 0 ) does not have a valid a, b-ranking. By Lemma 3.14, we conclude that rot−i (π) does not have a valid a, b-ranking. By the last paragraph, we may assume that for every block B ∈ krew(π) and every index i ∈ B −{max(B)}, we have (i, max(B)) ∈ A(a, b). Since π ∈ NC(b−1)−NC(a, b), by Proposition 3.3 there exists a block B ∈ krew(π) and indices i, j ∈ B − {max(B)} with i < j such that l am a am l (k − i) − (k − j) < (j − i) , b b b where max(B) = k. The set partition π refines the three-block noncrossing partition π 0 := {[i + 1, j], [j + 1, k], [b − 1] − [i + 1, k]}. Therefore, the set partition rot−i (π) refines rot−i (π 0 ) = {[1, j − i], [j − i + 1, k − i], [k − i + 1, b − 1]}. Since rot−i (π 0 ) consists of three intervals, we have that rot−i (π 0 ) can be obtained from rot−i (π) by a sequence of coarsenings as in Lemma 3.14. We show that rot−i (π 0 ) does not have a valid a, b-ranking; by Lemma 3.14, this implies that rot−i (π) does not have a valid a, b-ranking and completes the proof. Working towards a contradiction, suppose π 00 := rot−i (π 0 ) has a valid a, b-ranking. Denote the blocks of π 00 by B1 = [1, j − i], B2 = [j − i + 1, k − i], and B3 = [k − i + 1, b − 1]. We have 00 00 π 00 (B3 ) = a. On the other hand, by Lemma 3.14, we have that rankπa,b (B1 ) + rankπa,b (B2 ) + ranka,b 000 000 that π := {B1 ∪ B2 , B3 } also has a valid a, b-ranking. Moreover, we have that rankπa,b (B3 ) = 00 rankπa,b (B3 ). This implies that l am 000 000 00 a = rankπa,b (B1 ∪ B2 ) + rankπ (B3 ) = (k − i) + rankπ (B3 ). b

14


Putting these facts together gives 00

00

00

a = rankπa,b (B1 ) + rankπa,b (B2 ) + rankπa,b (B3 ) l am l am 00 = (j − i) + (k − j) + rankπ (B3 ) b b l l am am am l = (j − i) + (k − j) + a − (k − i) b b l b am a > a + (j − i) − (j − i) b b > a, which is a contradiction. We conclude that π 00 does not have a valid a, b-ranking.

As an application of Proposition 3.15, we get that the set NC(a, b) carries an action of not just the rotation operator rot, but the full dihedral group hrot, rfni. This gives another application of a, b-ranks. Corollary 3.16. Let a < b be coprime. The set NC(a, b) of a, b-noncrossing partitions is closed under the action of the dihedral group hrot, rfni. Proof. It is enough to show that NC(a, b) is closed under rfn. For any noncrossing partition π ∈ NC(b − 1) and any block B of π, we have that rfn(π)


(rfn(B)).

It follows that every element in the rotation orbit of π has a valid a, b-ranking if and only if every element in the rotation orbit of rfn(π) has a valid a, b-ranking. 4. Modified rank sequences In this section we will study rational noncrossing partitions which have nontrivial rotational symmetry. Our key tool will the the theory of a, b-rank developed in Section 3. We fix the following Notation. For the remainder of this section, let a < b be coprime positive integers. Let d|(b − 1) be a divisor with 1 ≤ d < b − 1. Let q := b−1 d . Let NCd (a, b) := {π ∈ NC(a, b) : rotd (π) = π} denote the set of a, b-noncrossing partitions which are fixed by rotd . The numerology of NCd (a, b) will turn out to be somewhat simpler than that of NC(a, b) itself. We begin by defining a modified version of the rank sequence which is well suited to studying partitions with rotational symmetry. Let π ∈ NCd (a, b). The fact that π is noncrossing implies that π has at most one block B0 which satisfies rotd (B0 ) = B0 . If such a block B0 exists, it is called central. Moreover, the cyclic group Zq = hrotd i acts freely on the non-central blocks of π. A non-central block B of π is called wrapping if the interval [min(B), max(B)] contains every block in the hrotd i-orbit of B. Any hrotd i-orbit of blocks has at most one wrapping block. Definition 4.1. Let π ∈ NCd (a, b). The d-modified rank sequence of π is the length d sequence Sd (π) = (s1 , . . . , sd ) of nonnegative integers defined by ( rankπa,b (B) if i = min(B) for some non-central, non-wrapping block B ∈ π, (13) si := 0 otherwise. For example, suppose that (a, b) = (4, 9), d = 4, and π = {{1, 8}, {2, 3, 6, 7}, {4, 5}} ∈ NC4 (4, 9). The block {1, 8} of π is wrapping and the block {2, 3, 6, 7} of π is central. Since the 4, 9-rank of {4, 5} is 1, the 4-modified rank sequence of π is S4 (π) = (0, 0, 0, 1).


15

The d-modified rank sequence Sd (π) is like the ordinary rank sequence R(π), but we only consider the indices in [d] rather than in [b−1] and we only keep track of the ranks of blocks which are neither central nor wrapping. It is true, but not obvious at this point, that a set partition π ∈ NCd (a, b) is determined by Sd (π). Our first lemma states that the assignment π 7→ Sd (π) commutes with the action of rotation. Lemma 4.2. Let π ∈ NCd (a, b) and let Sd (π) = (s1 , s2 , . . . , sd ) be the d-modified rank sequence of π. We have that (14)

Sd (rot(π)) = rot(Sd (π)),

where rot(s1 , s2 , . . . , sd ) = (sd , s1 , . . . , sd−1 ) is the rotation operator on sequences. Proof. Let Sd (rot(π)) = (s01 , s02 , . . . , s0d ) be the d-modified rank sequence of rot(π) and let 1 ≤ i ≤ q. We show that s0i = si−1 , where subscripts are interpreted modulo d. We will make free use of rot(π) Proposition 3.10, which implies that rankπa,b (B) = ranka,b (rot(B)) for any block B ∈ π. Case 1: 2 ≤ i ≤ d. Suppose si−1 > 0. Then i − 1 = min(B) for some non-central, non-wrapping block B ∈ π. Since B is non-central and non-wrapping and 1 ≤ min(B) ≤ d − 1, we know that rot(B) is also non-central and non-wrapping with min(rot(B)) = i. We conclude that s0i = si−1 . Suppose si−1 = 0. If i − 1 is not the minimum element of a block of π, then i is not the minimum element of a block of rot(π), so that s0i = 0. If i − 1 = min(B0 ) for a central block B0 ∈ π, then rot(B0 ) is a central block in rot(π) with i ∈ rot(B0 ), so that s0i = 0. If i − 1 = min(B) for a wrapping block B ∈ π, then the fact that 1 ≤ min(B) ≤ d − 1 implies that either i 6= min(rot(B)) or (rot(B) is wrapping with i ∈ rot(B)). In either situation, we get that s0i = 0. Case 2: i = 1. Suppose sd > 0. Then d = min(B) for some non-central, non-wrapping block B ∈ π. Recalling that rotd (π) = π, it follows that rotd(q−1)+1 (B) is a non-central, non-wrapping rot(π) block of rot(π) containing 1. Thus, we get s01 = ranka,b (rotd(q−1)+1 (B)) = rankπa,b (B) = sq . Suppose sd = 0. If d is contained in a central block of π, then 1 is contained in a central block of rot(π) and s01 = 0. Since π is noncrossing and rotd (π) = π, the index q cannot be contained in a wrapping block of π. If d ∈ B for some block B ∈ π which is non-central and non-wrapping, we must have that d 6= min(B). Since π is noncrossing with rotd (π) = π, it follows that rot(B) is wrapping and 1 ∈ rot(B), so that s01 = 0. What sequences (s1 , . . . , sd ) of nonnegative integers arise as d-modified rank sequences of partitions in NCd (a, b)? If π ∈ NCd (a, b) and Sd (π) = (s1 , . . . , sd ) is the d-modified rank sequence of π, we claim that X q(s1 + · · · + sd ) = rankπa,b (B), B

where the sum is over all non-central blocks B ∈ π. Indeed, each q-element orbit of non-central blocks contributes the rank of one of its constituents precisely once to the nonzero terms in Sd (π). By Proposition 3.15, a s1 + · · · + sd ≤ , q with equality if and only if π does not have a central block. (Unless q|a, the partition π necessarily has a central block.) We call a length d sequence (s1 , . . . , sd ) of nonnegative integers good if we have the inequality s1 + · · · + sd ≤ aq . The goal for the remainder of this section is to show that the map Sd : NCd (a, b) −→ {good sequences (s1 , . . . , sd )} is a bijection. Since good sequences are easily enumerated, this will give us information about NCd (a, b). The strategy is to isolate nice subsets of NCd (a, b) and the set of good sequences which contain at least one representative from every rotation orbit, show that these subsets are in bijection under Sd , and apply Lemma 4.2.

16


Definition 4.3. A set partition π ∈ NCd (a, b) is noble if π does not contain any wrapping blocks and, if π contains a central block B0 , we have that 1 ∈ B0 . For example, consider the case (a, b) = (6, 7) and d = 3. We have that π = {{1}, {2, 3, 5, 6}, {4}} ∈ NC3 (6, 7) is not a noble partition because 1 is not contained in the central block. On the other hand, both of the rotations {{1, 3, 4, 6}, {2}, {5}} and {{1, 2, 4, 5}, {3}, {6}} of π are noble partitions. Also, if (a, b) = (6, 7) and d = 2 we have that σ = {{1, 6}, {2, 3}, {4, 5}} ∈ NC2 (6, 7) is not a noble partition because the block {1, 6} is wrapping. However, the rotation {{1, 2}, {3, 4}, {5, 6}} of σ is a noble partition. In general, we have the following observation. Observation 4.4. Every hroti-orbit in NCd (a, b) contains at least one noble partition. The notion of nobility for sequences involves an intermediate definition. Let s = (s1 , . . . , sd ) be a good sequence. We call s very good if s1 + · · · + sd = aq or s1 = 0. We define a map L : {very good sequences} −→ {lattice paths from (0, 0) to (b, a)} as follows. If s = (s1 , . . . , sq ) is a very good sequence, let L(s) be the lattice path obtained by taking the q-fold concatenation of the path N s1 E . . . N sd E, adding a terminal east step, and if s1 + · · · + sd < aq adding an initial vertical run of size c := a − q(s1 + · · · + sd ). In symbols, ( if s1 + · · · + sd = aq , (N s1 E . . . N sd E)q E L(s) := (N c EN s2 E . . . N sd E)(N s1 E . . . N sd E)q−1 E if s1 + · · · + sd < aq . Since s is assumed to be very good, we get that L(s) ends at (b, a) so that the map L is well defined. We will refer to the subpaths L1 , . . . , Lq defined by the above factorization of L(s) as the segments of L(s), so that L(s) = L1 · · · Lq E. If s is a very good sequence, the lattice path L(s) is typically not an a, b-Dyck path. For example, if (a, b) = (4, 7), d = 3, and s = (s1 , s2 , s3 ) = (0, 1, 1), we have that L(s) = (N 0 EN 1 EN 1 E)(N 0 EN 1 EN 1 E)E, which is not a 4, 7-Dyck path. Definition 4.5. A very good sequence s = (s1 , . . . , sd ) is noble if L(s) is an a, b-Dyck path. When (a, b) = (4, 7) and d = 3, the above example shows that (0, 1, 1) is not a noble sequence. On the other hand, the rotation (1, 1, 0) of (0, 1, 1) is a noble sequence. In general, we have the following analog of Observation 4.4 for good sequences. Lemma 4.6. Every good sequence is hroti-conjugate to at least one noble sequence. Proof. Let (s1 , . . . , sd ) be a good sequence and set c = a − q(s1 + · · · + sd ). Case 1: c = a. In this case (s1 , . . . , sd ) is the zero sequence (0, 0, . . . , 0) and is trivially noble. Case 2: 0 < c < a. The argument we present here is a modification of the argument used to prove the Cycle Lemma. Consider the lattice path L which starts at the origin and ends at (2d, 2(s1 + · · · + sd )) given by L = (N s1 E . . . N sd E)(N s1 E . . . N sd E). As is common in rational Catalan theory, we label the lattice points P on L with integers w(P ) as follows. The origin is labeled 0. Reading L from left to right, if P and P 0 are consecutive lattice points, we set w(P 0 ) = w(P ) − a if P 0 is connected to P with an E-step and w(P 0 ) = w(P ) + b if P 0 is connected to P with an N -step. For example, suppose that (a, b) = (11, 13), d = 4, and (s1 , s2 , s3 , s4 ) = (1, 0, 2, 0). We have that q = 13 4 = 3 and c = 11 − 3(1 + 0 + 2 + 0) = 2. The lattice path L, together with the labels of its lattice points, is as follows.

CYCLIC SIEVING AND RATIONAL CATALAN THEORY 12

8 17 4

13 2

6

-3

1

17

-10

-1 -14

P0

-5

-9

0

By coprimality, there exists a unique lattice point on L of minimal weight. Let P0 be this lattice point; in the above example, we have w(P0 ) = −14. We claim that P0 occurs after a pair of consecutive east steps EE. Indeed, since 0 < c < a, we know that the weight of the terminal lattice point of L is negative (in the above example, −10), so that P0 is not the origin. If P0 did not occur after a pair of consecutive east steps, by the minimality of w(P0 ), we have that P0 occurs after a pair N E. But since a < b the lattice point P00 occurring at the beginning of this N E-sequence satisfies w(P00 ) = w(P0 ) + a − b < w(P0 ), a contradiction. Therefore, the lattice point P0 does indeed occur after a pair of consecutive east steps EE. The minimality of w(P0 ) and the fact that (s1 , . . . , sd ) 6= (0, . . . , 0) mean that P0 is immediately followed by a nonempty vertical run N si for some 1 ≤ i ≤ d. Since P0 is preceded by a pair EE, we get that i ≥ 2 and si−1 = 0. In the above example, we have that i = 3. Since si−1 = 0, the rotated sequence (si−1 , si , . . . , sd , s1 , s2 , . . . , si−2 ) is very good. The lattice path L(si−1 , si , . . . , sd , s1 , s2 , . . . , si−2 ) is therefore well defined. In the above example, we have (si−1 , si , . . . , sd , s1 , s2 , . . . , si−2 ) = (0, 2, 0, 1) and the lattice path L(0, 2, 0, 1) is shown in the proof of Lemma 4.7 below. The segmentation L(s) = L1 . . . Lq E = L1 L2 L3 E is shown with vertical hash marks. Observe that L(s) is an a, b-Dyck path in this case. We claim that the lattice path L(si−1 , si , . . . , sd , s1 , s2 , . . . , si−2 ) is always an a, b-Dyck path, so that the rotation (si−1 , si , . . . , sd , s1 , s2 , . . . , si−2 ) is a noble sequence. Indeed, consider the segmentation L(s) = L1 . . . Lq E. The segments L1 , . . . , Lq will be progressively further east, so it is enough to show that the final segment Lq stays west of the line y = x. By construction, the segment Lq starts with a single east step, hits the copy of the point P0 , then then has a nonempty vertical run, and eventually ends at the point (b − 1, a). Since (s1 , . . . , sd ) is a good sequence, we know that Lq starts at a lattice point to the west of the line y = ab (x + 1). The minimality of w(P0 ) forces Lq to remain west of the line y = ab x. Case 3: c = 0. This is a special case of the Cycle Lemma; the argument is very similar to Case 2 and is only sketched. We again consider the lattice path L given by L = (N s1 E . . . N sd E)(N s1 E . . . N sd E) and assign weights w(P ) to the lattice points P on L as before. There exists a unique lattice point P0 on L with minimal weight. The lattice point P0 necessarily occurs before a nonempty vertical run N si for some 1 ≤ i ≤ d. The rotation (si , si+1 , . . . , sd , s1 , s2 , . . . , si−1 ) of s is a noble sequence. Given any noble sequence s, we may consider the a, b-noncrossing partition π(L(s)) corresponding to the Dyck path L(s). We prove that π(L(s)) is rotd -invariant and noble. Lemma 4.7. Suppose that s = (s1 , . . . , sd ) is a noble sequence Then π := π(L(s)) is a noble a, b-noncrossing partition (and in particular rotd (π) = π) with Sd (π) = s. Proof. Recall the visibility bijection between blocks of π and nonempty vertical runs in L(s). Let c = a − q(s1 + · · · + sd ). The argument depends on whether c > 0 or c = 0. Case 1: c > 0. We consider the segmentation of L(s) given by L(s) = L1 L2 . . . Lq E,

18


where L1 = N c EN s2 E . . . N sd E and Lj = N s1 EN s2 E . . . N sd E for 2 ≤ j ≤ q. As an example of this case, consider (a, b) = (11, 13), d = 4, s = (0, 2, 0, 1). Then s is a noble sequence. The 11, 13-Dyck path L(s) = L1 L2 L3 E is shown below, where vertical hash marks separate the segments L1 , L2 , and L3 . The valley lasers from L(s) are as shown. For visibility, we suppress the labels on the interior lattice points.

Since s is noble and c > 0, we have s1 = 0. Fix any index 2 ≤ i ≤ d such that si > 0 and another index 1 ≤ j ≤ q − 1. Both of the segments Lj and Lj+1 of the a, b-Dyck path L(s) contain a copy of the nonempty vertical run N si . If P0 and P1 are the valleys at the bottom of these vertical runs in Lj and Lj+1 , respectively, we have that the lasers `(P0 ) and `(P1 ) fired from P0 and P1 are (rigid) translations of the same line segment. In particular, the block visible from the copy of N si in Lj+1 is the image of the block visible from the copy of N si in Lj by the operator rotd . Moreover, the fact that s1 = 0 means that neither of these blocks contain the index 1. In the above example, we have three segments (L1 , L2 , and L3 ) with si > 0 for i = 2, 4. The blocks visible from the copies of N s2 = N 2 in L1 , L2 , and L3 are {2, 3}, {6, 7}, and {10, 11}, respectively. The blocks visible from the copies of N s4 = N 1 in L1 , L2 , and L3 are {4}, {8}, and {12}, respectively. The block visible from the initial vertical run is {1, 5, 9}, which is central. In general, we conclude that the set of blocks of π not containing 1 is stable under the action of rotd , so that the block of π containing 1 must be central. We get that π ∈ NCd (a, b). Since π ∈ NCd (a, b) has a central block containing 1, the partition π contains no wrapping blocks. It follows that π in noble and Sd (π) = s. Case 2: c = 0. In this case, π does not contain a central block. We again consider the segmentation L(s) = L1 L2 . . . Lq E as in Case 1. Here Lj = N s1 EN s2 E . . . N sd E for 1 ≤ j ≤ q. As an example of this case, consider (a, b) = (9, 13), d = 4, and s = (1, 2, 0, 0). We have that s is a noble sequence. The 9, 13-Dyck path L(s) is shown below, with diagonal hash marks denoting the segmentation L(s) = L1 L2 L3 E. The valley lasers of L(s) are shown, and the interior lattice point labels are suppressed.

Let 1 ≤ i ≤ d be an index such that si > 0 and consider any two consecutive segments Lj and Lj+1 of L(s). The lasers fired from the valleys below the copies of N si both Lj and Lj+1 are either


19

• translates of each other, or • both hit the Dyck path L(s) = L1 . . . Lq E on its terminal east step. In the above example, we see that the lasers fired from each copy of N s2 = N 2 are translates of one another, and the lasers fired from each copy of N s1 = N 1 hit L(s) on its terminal east step. This implies that the blocks corresponding of these vertical runs are hrotd i-conjugate, so that π ∈ NCd (a, b). Moreover, since c = 0 the laser fired from any copy of the (nonempty) vertical run N s1 in L(s) = L1 . . . Lq E must hit L(s) on its terminal east step. This implies that π has no wrapping blocks, so that π is noble and Sd (π) = s. We show that nobility for sequences and nobility for partitions coincide. Lemma 4.8. Suppose that π ∈ NCd (a, b). Then π is noble if and only if Sd (π) is noble. Proof. Write Sd (π) = (s1 , . . . , sd ) and set c = a − q(s1 + · · · + sd ). (⇒) Suppose π is noble. Since π contains no wrapping blocks, the rank sequence R(π) is given by the formula ( (s1 , . . . , sd , s1 , . . . , sd , . . . , s1 , . . . , sd ) if s1 + · · · + sd = aq , R(π) = (c, . . . , sd , s1 , . . . , sd , . . . , s1 , . . . , sd ) if s1 + · · · + sd < aq . If c > 0, since 1 is contained in the central block of π we get s1 = 0, so that Sd (π) is very good. By Proposition 3.8, we get that L(s) is an a, b-Dyck path, so that Sd (π) is noble. (⇐) Suppose Sd (π) is noble. We claim that π contains no wrapping blocks. Working towards a contradiction, assume that π contains at least one wrapping block and choose a wrapping block B of π such that the interval [min(B), max(B)] is maximal under containment. If 1 ∈ B, then we would have s1 = 0 since B is wrapping, making the first step of the path L(Sd (π)) an east step. This contradicts the nobility of Sd (π). We conclude that 1 ∈ / B. Since 1 ∈ / B, the interval [1, min(B) − 1] is nonempty. By our choice of B, we have that [1, min(B) − 1] is a union blocks of π, none of which are central or wrapping. This means that the first min(B) − 1 terms in the d-modified rank sequence Sd (π) coincide with the first min(B) − 1 terms in the ordinary rank sequence R(π). By Proposition 3.8, the a, b-Dyck path corresponding to π starts at the origin with the subpath s N 1 EN s2 E . . . N smin(B)−1 E. Since [1, min(B) − 1] isa union of blocks of π, Corollary 3.9 guarantees that this subpath ends at the point (min(B) − 1, ab (min(B) − 1) ). It follows that the subpath (N s1 EN s2 E . . . N smin(B)−1 E)E obtained by appending a single east step crosses the diagonal y = ab x. But since B is wrapping, we have smin(B) = 0, so that this is an initial subpath of L(Sd (π)), so that L(Sd (π)) is not an a, b-Dyck path. This contradicts the nobility of Sd (π). We conclude that π contains no wrapping blocks. Suppose that π contains a central block B0 . We need to prove that 1 ∈ B0 . If 1 ∈ / B0 , the fact that π does not contain wrapping blocks implies that [1, min(B0 ) − 1] is a union of nonwrapping, non-central blocks of π. The first min(B0 )−1 terms of the d-modifed rank sequence Sd (π) coincide with the corresponding terms of the ordinary rank sequence R(π). The same reasoning as in the last paragraph implies that the lattice path L(Sd (π)) contains the point (min(B0 ) − 1, ab (min(B0 ) − 1) ). However, since B0 is central, we have that smin(B0 ) = 0, so that the lattice path L(Sd (π)) has an east step originating from this lattice point. But this means that L(Sd (π)) is not an a, b-Dyck path, contradicting the nobility of Sd (π). We conclude that 1 ∈ B0 , and that π is a noble partition. We have the lemmata we need to prove that the map Sd is a bijection. In particular, partitions in NCd (a, b) are determined by their d-modified rank sequences.

20


Proposition 4.9. The map Sd : NCd (a, b) −→ {good sequences (s1 , . . . , sd )} is a bijection which commutes with the action of rotation. Proof. By Lemma 4.2, we know that Sd commutes with rotation. If s is a noble sequence, Lemma 4.7 shows that Sd−1 (s) is nonempty, and Lemma 4.6 shows that Sd is surjective. To see that Sd is injective, let s be a noble sequence and suppose π ∈ NCd (a, b) satisfies Sd (π) = s = (s1 , . . . , sd ). By Lemma 4.8, the partition π is noble. The rank sequence R(π) is therefore ( (s1 , . . . , sd , s1 , . . . , sd , . . . , s1 , . . . , sd ) if s1 + · · · + sd = aq , R(π) = (c, . . . , sd , s1 , . . . , sd , . . . , s1 , . . . , sd ) if s1 + · · · + sd < aq , where c = a − q(s1 + · · · + sd ). By Proposition 3.8, any a, b-noncrossing partition is determined by its rank sequence. We conclude that |Sd−1 (s)| = 1. By Observation 4.4 and Lemma 4.6, together with the fact that Sd commutes with rot (Lemma 4.2), we conclude that Sd is a bijection. Since every nonzero entry in a d-modified rank sequence Sd (π) = (s1 , . . . , sd ) corresponds to a hrotd i-orbit of non-central blocks of π of that rank, the following result follows immediately from Proposition 4.9. Corollary 4.10. Let a < b be coprime positive integers and let d|(b − 1) be a divisor with 1 ≤ d < b − 1. Let m1 , . . . , ma be nonnegative integers which satisfy b−1 d (m1 + 2m2 · · · + ama ) ≤ a. Write m := m1 + m2 + · · · + ma . The number of a, b-noncrossing partitions which are invariant under rotd and have mi orbits of non-central blocks of a, b-rank i under the action of hrotd i is the multinomial coefficient d . m1 , m2 , . . . , ma , d − m The Fuss-Catalan case b ≡ 1 (mod a) of Corollary 4.10 is equivalent to a result of Athanasiadis [3, Theorem 2.3]. We can also consider counting partitions in NCd (a, b) with a fixed number of non-central block orbits (of any rank). By Proposition 4.9, this is equivalent to counting sequences (s1 , . . . , sd ) of nonnegative integers with bounded sum and a fixed number of nonzero entries. Corollary 4.11. Let a < b be coprime positive integers and let d|(b − 1) be a divisor with 1 ≤ d < b − 1. Let p be a nonnegative integer with b−1 d p ≤ a. The number of a, b-noncrossing partitions which are invariant under rotd , have a central block, and have p orbits of non-central blocks under the action of hrotd i is ad d b b−1 c − 1 . p p The number of a, b-noncrossing partitions which are invariant under rotd , do not have a central block, and have p orbits of non-central blocks under the action of hrotd i is ( ad d b−1 −1 if b−1 d | a, p p−1 0

if

b−1 d

- a.

Proof. For the first part, we choose p entries in the sequence (s1 , . . . , sd ) to j be knonzero. Then, we ad assign positive values to these p entries in such a way that their sum is < b−1 . The second part is similar. Finally, we can consider the problem of counting NCd (a, b) itself. By Proposition 4.9, thisjcorrek ad sponds to counting sequences (s1 , . . . , sd ) of nonnegative integers which satisfy s1 +· · ·+sd ≤ b−1 .


21

Corollary 4.12. Let a < b be coprime positive integers and let d|(b − 1) be a divisor with 1 ≤ d < b − 1. The number of a, b-noncrossing partitions which are invariant under rotd is

ad b b−1 c+d . d

The Fuss-Catalan cases b ≡ 1 (mod a) of Corollaries 4.11 and 4.12 are results of Reiner [10, Propositions 6 and 7].

5. Cyclic sieving Let X be a finite set, let C = hci be a finite cyclic group acting on X, let X(q) ∈ N[q] be a polynomial with nonnegative integer coefficients, and let ζ ∈ C be a root of unity with multiplicative order |C|. The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon (CSP) if, for all d ≥ 0, d we have X(ζ d ) = |X c | = |{x ∈ X : cd .x = x}| (see [12]). In this section we prove cyclic sieving results for the action of rotation on NC(a, b). Our proofs will be ‘brute force’ and use direct root-of-unity evaluations of q-analogs. We will make frequent use of the following fact: If x ≡ y (mod z), then

lim q→e2πi/z

[x]q = [y]q

(

x y

0

if y ≡ 0 mod z, otherwise.

From this, we get the useful fact that lim q→e2πi/z

[nz]q ! = [kz]q !

n [nz − kz]q !|q=e2πi/z . k

Theorem 5.1. LetPa < b be coprime and Plet r = (r1 , r2 , . . . , ra ) be a sequence of nonnegative integers satisfying ai=1 iri = a. Set k := ai=1 ri . Let X be the set of a, b-noncrossing partitions of [b − 1] with r1 blocks of rank 1, r2 blocks of rank 2, . . . , and ra blocks of rank a. The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon, where C = Zb−1 acts on X by rotation and (15)

X(q) = Krewq (a, b, r) =

[b − 1]!q [r1 ]!q [r2 ]!q · · · [ra ]!q [b − k]!q

is the q-rational Kreweras number. Proof. Reiner and Sommers proved that the q-Kreweras number Krewq (a, b, r) is polynomial in q with nonnegative integer coefficients using algebraic techniques [11]. No combinatorial proof of the polynomiality or the positivity of Krewq (a, b, r) is known. 2πi

d Let ζ = e b−1 and let d|(b − 1) with 1 ≤ d < b − 1. Write t = b−1 d . We have that X(ζ ) = 0 unless at t|ri for all but at most one 1 ≤ i ≤ a, and that ri0 ≡ 1 (mod t) if t - ri0 . If the sequence r satisfies the condition of the last sentence, define a new sequence (m1 , . . . , ma ) by mi = rti for 1 ≤ i ≤ a. Let m = m1 + · · · + ma . Write ri0 = ci0 t + si0 for si0 ∈ {0, 1} and assume t|ri for all i 6= i0 . We have

22


[b − 1 − mt]!q [ri0 − si0 ]!q d − (m − ma ) d − m1 ··· lim lim X(q) = m2 ma [ri0 ]!q [b − k]!q q→ζ d q→ζ d [b − 1 − (k − si0 )]!q [ri0 − si0 ]!q d b = lim m1 , . . . , ma , d − m q→ζ d [ri0 ]!q [b − k]!q ( d 1 s i0 = 0 m1 ,...,ma ,d−m limq→ζ d [b−k]q = d 1 s i0 = 1 m1 ,...,ma ,d−m limq→ζ d [ri0 ]q d = . m1 , . . . , ma , d − m

d m1

d

By Corollary 4.10, we have X(ζ d ) = |X rot |.

The following Narayana version of Theorem 5.1 proves a CSP involving the action of rotation on a, b-noncrossing partitions with a fixed number of blocks. Theorem 5.2. Let a < b be coprime, let 1 ≤ k ≤ a, and let X be the set of (a, b)-noncrossing partitions of [b − 1] with k blocks. The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon, where C = Zb−1 acts on X by rotation and 1 a b−1 (16) X(q) = Narq (a, b, k) = [a]q k q k − 1 q is the q-rational Narayana number. Proof. Reiner and Sommers proved that the q-Narayana numbers Narq (a, b, k) are polynomials in q with nonnegative integer coefficients [11]. As in the Kreweras case, no combinatorial proof of this fact is known. 2πi Let ζ = e b−1 and let d|(b − 1) with 1 ≤ d < b − 1. Let q = b−1 d . By Corollary 4.11, it is enough to show that  ad  d b b−1 c−1  if k ≡ 0 (mod q),  b kd c  b kq c−1  ad d b b−1 c X(ζ d ) = if k ≡ 1 (mod q), k  b c b kq c  q   0 otherwise. The argument here is similar to that in the proof of Theorem 5.1 and is left to the reader. The next CSP was asked for in [2, Subsection 6.2]. Theorem 5.3. Let a < b be coprime and let X be the set of (a, b)-noncrossing partitions of [b − 1]. The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon, where C = Zb−1 acts on X by rotation and 1 a+b (17) X(q) = Catq (a, b) = [a + b]q a, b q is the q-rational Catalan number. 2πi

Proof. Let ζ = e b−1 and let d|(b − 1) with 1 ≤ d < b − 1. By Corollary 4.12 it is enough to show that ad b b−1 c + d X(q) = . d The argument here is similar to that in the proof of Theorem 5.1 and is left to the reader. Our final CSP proves [2, Conjecture 5.3].

CYCLIC SIEVING AND RATIONAL CATALAN THEORY 4

4 1 2

2 5 3

1

5

6

23

1

7

2

7 2

3,5 3 6

3 1,4 5

4

Figure 5. A 5, 8-noncrossing parking function. Theorem 5.4. Let (a, b) be coprime and let X be the set of homogeneous (a, b)-noncrossing partitions of [a + b − 1]. The triple (X, C, X(q)) exhibits the cyclic sieving phenomenon, where C = Za+b−1 acts on X by rotation and a+b 1 (18) X(q) = Catq (a, b) = [a + b]q a, b q is the q-rational Catalan number. Proof. Let D be an a, b-Dyck path. Construct an a, a + b-Dyck path D0 by replacing every north step in D with a pair N E. Then D0 has a vertical runs and π(D) = π(D0 ). Moreover, the map D 7→ D0 gives a bijection {all a, b-Dyck paths} −→ {all a, a + b-Dyck paths with a vertical runs}. It follows that the set HNC(a, b) of homogeneous a, b-noncrossing partitions of [a + b − 1] is precisely the set of (ordinary) a, a + b-noncrossing partitions of [a + b − 1] with precisely a blocks. The result follows from Theorem 5.2. 6. Parking functions 6.1. Noncrossing parking functions. Let W be an irreducible real reflection group with Coxeter C number h. Armstrong, Reiner, and Rhoades defined a W × Zh -set ParkN W called the set of W C noncrossing parking functions [1]. Given a Fuss parameter m ≥ 1, a Fuss extension ParkN W (m) of C ParkN W was defined in [14]. An increasingly strong trio of conjectures (Weak, Intermediate, and Strong) was formulated about these objects and it was shown that the weakest of these uniformly implies various facts from W -Catalan theory which are at present only understood in a case-by-case fashion. In this section, we give a rational extension of the constructions in [1, 14] when W = Sa is the symmetric group. This gives evidence that NC(a, b) gives the ‘correct’ definition of rational noncrossing partitions. Extending the work of [1, 14] to other reflection groups remains an open problem. Definition 6.1. Let a < b be coprime. An a, b-noncrossing parking function is a pair (π, f ) where π ∈ NC(a, b) is an a, b-noncrossing partition and B 7→ f (B) is a labeling of the blocks of π with subsets of [a] such that U • we have [a] = B∈π f (B), and • for all blocks B ∈ π we have |f (B)| = rankπa,b (B). We denote by ParkN C (a, b) the set of all a, b-noncrossing parking functions.

24


An example of a 5, 8-noncrossing parking function is shown in Figure 5. The parking function shown is (π, f ), where π = {{1, 3, 7}, {2}, {4, 5, 6}} and f is the labeling {1, 3, 7} 7→ {3, 5}, {4, 5, 6} 7→ {1, 4}, and {2} 7→ {2}. By Proposition 3.6, when (a, b) = (n, mn + 1), the set ParkN C (n, mn + 1) agrees with the NC C (n, n+ construction of ParkN Sn (m) given in [14]. In the classical case (a, b) = (n, n+1), the set Park 1) appeared in the work of Edelman under the name of ‘2-noncrossing partitions’ [6]. The set ParkN C (a, b) carries an action of Sa × Zb−1 . Proposition 6.2. The set ParkN C (a, b) carries an action of the product group Sa × Zb−1 , where Sa acts by label permutation and Zb−1 acts by rotation. Proof. We need to know that rotation preserves a, b-ranks of blocks, which is Proposition 3.10.

In order to state a formula for the character of the action in Proposition 6.2, we will need some 2πi notation. Let V = Ca /h(1, . . . , 1)i be the reflection representation of Sa and let ζ = e b−1 . Given w ∈ Sa and d ≥ 0, let multw (ζ d ) be the multiplicity of ζ d as an eigenvalue in the action of w on V . With this notation, a formula for the character χ is given by the following formula. Theorem 6.3. Let w ∈ Sa and let g be a generator of Zb−1 . We have that (19)

χ(w, g d ) = bmultw (ζ

d)

for all w ∈ Sa and d ≥ 0. The multiplicity multw (ζ d ) can be read off from the cycle structure of w. Namely, for d|b − 1 we have that ( #(cycles of w) − 1 if q = 1, d (20) multw (ζ ) = #(cycles of w of length divisible by q) otherwise, where q = b−1 d . When (a, b) = (n, n + 1), Proposition 6.3 was proven in [1, Section 8]. When (a, b) = (n, mn + 1), Proposition 6.3 is [14, Proposition 8.6]. The character formula of Equation 19 is a rational extension of the Weak Conjecture of [1, 14] when W = Sa is the symmetric group. The argument used to prove [14, Proposition 8.6] can be combined with the enumerative results of Section 4 to prove Theorem 6.3. We quickly illustrate how this is done. d

Proof. (of Theorem 6.3) Let (w, g d ) ∈ Sa × Zb−1 . We want to show that χ(w, g d ) = bmultw (ζ ) . Without loss of generality, we may assume that d|b − 1. Let q := b−1 d . The argument depends on whether q = 1 or q > 1. Case 1: q = 1. In this case, we are ignoring the action of Zb−1 and considering ParkN C (a, b) as an Sa -set. We construct an Sa -equivariant bijection from ParkN C (a, b) to another Sa -set which is known to have the correct character. Let Parka,b be the set of all sequences (p1 , p2 , . . . , pa ) of positive integers whose nondecreasing rearrangement (p01 ≤ p02 ≤ · · · ≤ p0a ) satisfies p0i ≤ ab (i − 1) + 1. Equivalently, the histogram with left-to-right heights (p01 − 1, p02 − 1, . . . , p0a − 1) stays below the line y = ab x. Sequences in Parka,b are called rational slope parking functions. The symmetric group Sa acts on Parka,b by w.(p1 , p2 , . . . , pa ) := (pw(1) , pw(2) , . . . , pw(a) ). It is known that the character of this action is given by Equation 19 with ζ = 1. ∼ We build an Sa -equivariant bijection ϕ : ParkN C (a, b) − → Parka,b as follows. Let (π, f ) be an a, b-noncrossing parking function. Define a sequence (p1 , p2 , . . . , pa ) by letting pi = min(B), where B is the unique block of π satisfying i ∈ f (B). Proposition 3.8 guarantees that (p1 , p2 , . . . , pa ) is a sequence in Parka,b , so that the assignment ϕ : (π, f ) 7→ (p1 , p2 , . . . , pa ) gives a well defined function ϕ : ParkN C (a, b) → Parka,b . It is clear that ϕ is Sa -equivariant. Moreover, if ϕ(π, f ) =


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(p1 , p2 , . . . , pa ), then we can recover both the minimal elements of the blocks of π (these are the entries appearing in p1 , p2 , . . . , pa ) and the ranks of these blocks (the rank of a block B is the number of times min(B) occurs in p1 , p2 , . . . , pa ). Proposition 3.8 says that the a, b-noncrossing partition π is therefore determined from (p1 , p2 , . . . , pa ). It is easy to see that we can determine the entire a, b-noncrossing function (π, f ), so that ϕ is an Sa -equivariant bijection. Case 2: q > 1. This argument is a rational extension of [14, Section 8]. Let rq (w) be the number of cycles of w having length divisible by q. We need to show that d

|ParkN C (a, b)(w,g ) | = brq (w) ,

(21) d

where ParkN C (a, b)(w,g ) is the set of a, b-noncrossing parking functions fixed by (w, g d ). The idea is to show that both sides of Equation 21 count a certain set of functions. Let g act on the set [b − 1] ∪ {0} by the permutation (1, 2, . . . , b − 1)(0). A function e : [a] → [b − 1] ∪ {0} is said to be (w, g d )-equivariant if we have e(w(j)) = g d e(j) for every 1 ≤ j ≤ a. We claim that the number of (w, g d )-equivariant functions e : [a] → [b−1]∪{0} is equal to brq (w) . Indeed, the values of e on any cycle of w are determined by the value of e on any representative of that cycle. Moreover, unless a cycle of w has length divisible by q, the relation e(w(j)) = g d e(j) forces e(j) = 0 for all j belonging to that cycle. For every cycle of w having length divisible by q, we have b choices for e(j), where j is a chosen cycle representative. By considering set partitions coming from preimages, we can get another expression for the number of (w, g d )-admissible functions [a] → [b − 1] ∪ {0}. A set partition σ = {B1 , B2 , . . . } of [a] is called (w, q)-admissible if • σ is w-stable in the sense that w(σ) = {w(B1 ), w(B2 ), . . . } = σ, • at most one block Bi0 of σ is itself w-stable in the sense that w(Bi0 ) = Bi0 , and • for any block Bi of σ which is not w-stable, the blocks Bi , w(Bi ), w2 (Bi ), . . . , wq−1 (Bi ) are pairwise distinct, and wq (Bi ) = Bi . It is straightforward to see that, for any (w, g d )-equivariant function e : [a] → [b − 1] ∪ {0}, the set partition σ of [a] defined by i ∼ j if and only if e(i) = e(j) is (w, q)-admissible. On the other hand, the same argument as in [14, Proof of Lemma 8.4] shows that the number of (w, g d )-equivariant functions e : [a] → [b − 1] ∪ {0} which induce a fixed (w, q)-admissible set partition σ of [a] is (b − 1)(b − 1 − q)(b − 1 − 2q) · · · (b − 1 − (tσ − 1)q), where tσ is the number of non-singleton w-orbits of blocks of σ. Combining this with the last paragraph, we get that X (22) brq (w) = (b − 1)(b − 1 − q)(b − 1 − 2q) · · · (b − 1 − (tσ − 1)q), σ

where the sum is over all (w, q)-admissible set partitions σ of [a]. To relate Equation 22 to parking functions, for (π, f ) ∈ ParkN C (a, b) we let τ (π, f ) be the set partition of [a] defined by i ∼ j if and only if i, j ∈ f (B) for some block B ∈ π. If (π, f ) ∈ d ParkN C (a, b)(w,g ) , it follows that τ (π, f ) is a (w, q)-admissible set partition of [a]. On the other hand, if σ is a fixed (w, q)-admissible partition of [a], we claim that the number of parking functions d (π, f ) ∈ ParkN C (a, b)(w,g ) with τ (π, f ) = σ equals (b − 1)(b − 1 − q)(b − 1 − 2q) · · · (b − 1 − (tσ − 1)q), where tσ is the number of nonsingleton w-orbits of blocks in σ. To see why this claim is true, we consider how to construct an a, b-noncrossing parking function d (π, f ) ∈ ParkN C (a, b)(w,g ) with τ (π, f ) = σ. This argument is almost the same as that proving [14, Lemma 8.5], but it will rely on Corollary 4.10. We first construct an a, b-noncrossing partition π which is invariant under rotd . If σ has mi non singleton w-orbits of blocks of size i for 1 ≤ i ≤ a, then π must have mi hrotd i-orbits of non-central blocks of rank i for 1 ≤ i ≤ a. Corollary 4.10 says that the number of such partitions π is the

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d . With π fixed, we consider how to build the labeling f of multinomial coefficient m1 ,m2 ,...,m a ,d−tσ the blocks of π. The labeling f must pair off the hrotd i-orbits of non-central blocks of π of rank i with the non-singleton w-orbits of blocks of σ of size i. For every i, there are mi ! ways to do this matching. As each of these orbits has size q, we also have q ways to rotate f within each orbit after d this matching is chosen. In summary, we have that the number of pairs (π, f ) ∈ ParkN C (a, b)(g,w ) satisfying τ (π, f ) = σ is d d! m1 ma q · · · q m1 ! · · · ma ! = q m1 · · · q ma m1 , . . . , ma , d − tσ (d − tσ )! = (b − 1)(b − 1 − q)(b − 1 − 2q) · · · (b − 1 − (tσ − 1)q). Applying Equation 22, we obtain Equation 21, completing the proof.

Theorem 6.3 can be strengthened to prove a rational analog of the Generic Strong Conjecture of L [15] in type A. Let V be the (complexified) reflection representation of Sa , let C[V ] = d≥0 C[V ]d be its polynomial ring, and equip C[V ] with the graded action of Sa × Zb−1 given by letting Sa act 2πi by linear substitutions and the generator g of Zb−1 scale by (e b−1 )d in degree d. We identify C[V ]1 with the dual space V ∗ and consider the set of C[Sa ]-equivariant linear maps HomC[Sa ] (V ∗ , C[V ]b ) as an affine complex space. We refer the reader to [15] for the definitions of the objects in the following result. Theorem 6.4. Let R ⊂ HomC[Sa ] (V ∗ , C[V ]b ) be the set of Θ ∈ HomC[Sa ] (V ∗ , C[V ]b ) such that the ‘parking locus’ V Θ (b) ⊂ V cut out by the ideal hΘ(x1 ) − x1 , . . . , Θ(xa−1 ) − xa−1 i ⊂ C[V ] is reduced (here x1 , . . . , xa−1 is any basis of V ∗ ). For any Θ ∈ R, there exists an equivariant bijection of Sa × Zb−1 -sets (23) V Θ (b) ∼ ParkN C (a, b). =S ×Z a

b−1

Moreover, there exists a nonempty Zariski open subset U ⊆ HomC[Sa ] (V ∗ , C[V ]b ) such that U ⊆ R. The proof of Theorem 6.4 is almost a word-for-word recreation of [15, Sections 4, 5]. One need only replace the reference to the proof of [14, Lemma 8.5] in the proof of [15, Lemma 4.6] with the corresponding argument the fifth paragraph of Case 2 in the proof of Theorem 6.3 (which ultimately relies on Corollary 4.10). 7. Closing Remarks This paper has focused entirely on rational Catalan theory for the symmetric group. The more ambitious problem of extending rational Catalan combinatorics to other reflection groups W is almost entirely open. However, the results of this paper give a roadmap for defining rational noncrossing partitions for the hyperoctohedral group. Let W (Bn ) denote the hyperoctohedral group of signed permutations of [n]. In the classical and Fuss-Catalan cases, objects associated to the group W (Bn ) are obtained by considering those attached to the ‘doubled’ symmetric group S2n which are invariant under antipodal symmetry. When (a, d) → (2n, b−1 2 ), the formulas in Corollaries 4.10, 4.11, and 4.12 reduce to the hyperoctohedral analogs of the rational Kreweras, Narayana, and Catalan numbers (here we view 2n as the Coxeter number of W (Bn ) and let the rational parameter b be coprime to 2n). Thus, restricting to objects with antipodal symmetry gives the correct numerology for type B, even in the rational setting. It would be interesting to see how far the techniques of this paper can be extended to develop on rational Catalan combinatorics outside of type A. 8. Acknowledgements The authors are grateful to Drew Armstrong and Vic Reiner for helpful conversations. B. Rhoades was partially supported by NSF grants DMS - 1068861 and DMS - 1500838.


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